Complex Arithmetic Operations Research Articles

Multiple Quadratic Forms: A Case Study in the Design of Data-Parallel Algorithms

Data-parallel implementations of the computationally intensive task of solving multiple quadratic forms (MQFs) have been examined. Coupled and uncoupled parallel methods are investigated, where coupling relates to the degree of interaction among the processors. Also, the impact of partitioning a large MQF problem into smaller non-interacting subtasks is studied. Trade-offs among the implementations for various data-size/machine-size ratios are categorized in terms of complex arithmetic operation counts, communication overhead, and memory storage requirements. Furthermore, the impact on performance of the mode of parallelism used is considered, specifically, SIMD versus MIMD versus SIMD/MIMD mixed-mode. From the complexity analyses, it is shown that none of the algorithms presented in this paper is best for all data-size/machine-size ratios. Thus, to achieve scalability (i.e., good performance as the number of processors available in a machine increases), instead of using a single algorithm, the approach discussed is to have a set of algorithms from which the most appropriate algorithm or combination of algorithms is selected based on the ratio calculated from the scaled machine size. The analytical results have been verified by experiments on the MasPar MP-1 (SIMD), nCUBE 2 (MIMD), and PASM (mixed-mode) prototype.

Event-based verification of synchronous, globally controlled, logic designs against signal flow graphs

We address the problem of automatically verifying large digital designs at the logic level, against high-level specifications. We present a technique which allows for the verification of a specific class of systems, namely systems with synchronous globally timed control. To a first approximation, these are systems where a single controller directs the data through the data path and decides (globally) when to move the data. We address the verification of these systems against a Signal Flow Graph (SFG) specification, or a specification in an applicative language such as SILAGE. In this paper, a method is presented for verifying the implementation against an intermediate SFG, which is an expansion of the original specification in such a way that all the operations correspond to Register Transfers (RT's) in the implementation. In this SFG, complex arithmetic operations such as multiplications may have been decomposed into simpler ones, such as shifts and additions, and new operations may have been introduced for maintaining iteration indices and computing addresses of memory locations. SFG's can be viewed as maximally parallel synchronous machines. Both the implementation and the specification are then Finite State Machines, having string functions (input/output mappings) associated with them. Correctness is taken to mean that a certain relation (the /spl beta/-relation) holds between these string functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Handling of Complex Numbers in the CHProgramming Language

The handling of complex numbers in the CHprogramming language will be described in this paper. Complex is a built-in data type in CH. The I/O, arithmetic and relational operations, and built-in mathematical functions are defined for both regular complex numbers and complex metanumbers of ComplexZero, Complexlnf, and ComplexNaN. Due to polymorphism, the syntax of complex arithmetic and relational operations and built-in mathematical functions are the same as those for real numbers. Besides polymorphism, the built-in mathematical functions are implemented with a variable number of arguments that greatly simplify computations of different branches of multiple-valued complex functions. The valid lvalues related to complex numbers are defined. Rationales for the design of complex features in CHare discussed from language design, implementation, and application points of views. Sample CHprograms show that a computer language that does not distinguish the sign of zeros in complex numbers can also handle the branch cuts of multiple-valued complex functions effectively so long as it is appropriately designed and implemented.

Application of Rader transforms to the analysis of nuclear spectral data

This paper describes a Rader transform method using a special arithmetic for the processing of nuclear spectral data. Rader transforms offer impressive computational savings vis-à-vis Fourier transform methods. Rader transforms require only integer additions and word shifts but no multiplications while Fourier transforms require complex arithmetic operations. Moreover, use of Rader transforms gives exact computations without any roundoff errors and does not require storage of basis functions. They are “the best transforms” for computer processing of nuclear spectral data. Rader transforms using a Fermat prime 65537 have been applied to deconvolve observed spectral data using a special filter function. A uniform improvement in resolution of 45% has been observed both in single and double spectral lines. A FORTRAN program GAMRAD is written to deconvolve spectral data using the special filter function.

A generalized object display processor architecture

A multiprocessor architecture has been developed which addresses the problem of the display and manipulation of multiple shaded three dimensional objects derived from emperical data on a raster scan CRT. Fully general control of such parameters as position, size, orientation, rotation, tone scale, and shading are accomplished at video rates permitting real-time interaction with the display presentation. The GODPA architecture is based on a large number of relatively simple processing elements which access their own memory modules without input conflict. Reconstruction algorithms are used which do not require any complex arithmetic or logical high speed operations. This hardware organization is highly modular and expandible and is ideally suited for implementation with VLSI technology.

IDA: An interactive data analysis system for clinical and experimental research

IDA is a general-purpose Interactive Data Analysis tool for small to medium size data bases, suitable for very small computers, as well as larger ones. It is aimed at users with little or no computer experience. IDA combines facilities for interactive data entry and editing, selective retrieval, and data analysis and presentation. All operations are controlled by the user through a simple and highly interactive dialogue. The user can specify complex logical relationships and arithmetic operations, and can request graphs, histograms, lists, tables and simple statistics.

A note on the "Evaluation of an analytical function of a companion matrix with distinct eigenvalues"

It was recently shown that companion matrices possess a recursive property which allows their function to be computed efficiently and accurately. In this letter, an additional recursive property of companion matrices is derived. These two properties are then combined to suggest a substantially improved algorithm for the evaluation of functions of companion matrices. It is also indicated that by using this algorithm, the complex arithmetic operations stemming from the presence of complex eigenvalues in real matrices are reduced to a bare minimum.